Bifurcation Trees Research Articles

Constructed limit cycles in a discontinuous system with multiple vector fields

In this paper, limit cycles in a discontinuous dynamical system with different vector fields in different domains are constructed. Such limit cycles are obtained through algebraic equations based on specific mapping structures. The stability and bifurcations of limit cycles are studied through eigenvalue analysis. The grazing and sliding bifurcations on the bifurcation trees are presented. Once a grazing or sliding bifurcation occurs, a limit cycle switches to a new limit cycle or vanishes. Such bifurcations (i.e., saddle-node bifurcation, Neimark bifurcation, period-doubling bifurcation, grazing bifurcation, and sliding bifurcation) are for onset and vanishing of limit cycles at specific parameters. This study presents how to construct limit cycles in discontinuous dynamical systems and how to develop the corresponding mathematical conditions of motion switchability at boundaries.

Independent Period-2 Motions to Chaos in a van der Pol–Duffing Oscillator

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

Abstract In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is presented for a better understanding of global behaviors an...

An algorithm for computing a stationary flow in a binary bifurcation tree

In this work we propose an algorithm for computing a stationary flow in a bifurcation tree. Our idea is to divide the tree into smaller basic blocks, each corresponding to one bifurcation, and solve a sequence of flow problems for the individual blocks. Numerical experiments demonstrate that the algorithm works well. We give a criteria for convergence that can be verified numerically and also an analytical convergence proof for an important special case.The application we have in mind is the computation of the time dependent blood flow in the arterial tree of the human body. The work presented here is for a simplified case but we discuss the extension of our work to the realistic cases. Also potential applications are discussed.

Bifurcation Trees of Period-1 to Period-2 Motions in a Periodically Excited Nonlinear Spring Pendulum

The spring-pendulum system has been of greet interest for a long time, and one tried to use the perturbation analysis to understand such a system. Until now, one cannot provide a satisfied result to explain the dynamics of the spring-pendulum system. In this paper, bifurcation trees of period-1 to period-2 motions in a periodically forced, nonlinear spring pendulum system are obtained through the discrete mapping method. The corresponding harmonic frequency-amplitude characteristics of period-1 to period-2 motions are presented, and the stability and bifurcations of period-1 to period-2 motions on the bifurcation trees are presented as well. From the analytical prediction, numerical illustrations of period-1 and period-2 motions are completed for comparison of numerical and analytical solutions. The results presented in this paper are totally different from the traditional perturbation analysis.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two Neimark bifurcations (NBs) of period-1 motions, one NB of period-2 motions, and four NBs of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling (PD) bifurcations exist for the period-1, period-2, and period-4 motions. The unstable PD bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable PD bifurcation is the third order source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

Periodic motions on bifurcation trees in an inverted pendulum with a periodically moving base

This paper studies periodic motions of an inverted pendulum with a periodically moving base. A bifurcation tree of period-1 to period-8 motions is predicted through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are obtained semi-analytically, and the corresponding stability and bifurcation of the periodic motions are determined by eigenvalue analysis. Numerical simulations are completed for illustration of motion complexity in the inverted pendulum with a periodically moving base. The numerical and analytical results are presented for comparison. Frequency–amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Through such results, one can better understand the building vibration during earthquake. In addition, stabilization of the inverted pendulum can be based on the periodically moving base.

Two Nonholonomic Chaotic Systems. Part I. On the Suslov Problem

A generalization of the Suslov problem with changing parameters is considered. The physical interpretation is a Chaplygin sleigh moving on a sphere. The problem is reduced to the study of a two-dimensional system describing the evolution of the angular velocity of a body. The system without viscous friction and the system with viscous friction are considered. Poincare maps are constructed, attractors and noncompact attracting trajectories are found. The presence of noncompact trajectories in the Poincare map suggests that acceleration is possible in this nonholonomic system. In the case of a system with viscous friction, a chart of dynamical regimes and a bifurcation tree are constructed to analyze the transition to chaos. The classical scenario of transition to chaos through a cascade of period doubling is shown, which may indicate attractors of Feigenbaum type.

Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

Parametric optimization of a frequency-up-conversion piezoelectric harvester via discontinuous analysis

In this study, the dynamical and electrical behaviors of an impact-based frequency-up-conversion energy harvester were studied based on discontinuous dynamics theory. This analytical study enables us to better understand the response of an impact-based frequency-up-conversion energy harvester as system parameters change, hence, guiding us to design a high-efficiency energy harvester via optimizing the values of the critical parameters of the system. For a given base excitation, the optimum gap to maximize the output power was obtained. The energy harvester consists of a sinusoidal vibrating piezoelectric bimorph and a stopper. The equations of the piezoelectric bimorph, which was modeled as an Euler–Bernoulli beam, were obtained based on the linear piezoelectric constitutive law. The generated voltage and power of the energy harvester were obtained via discontinuous dynamics analysis. Furthermore, the bifurcation diagrams of period-1 and period-2 motions were presented as the excitation frequency varying. To better understand the effect of different parameters on the performance of our system, the bifurcation trees of the period-1 motion versus varying excitation frequency were analytically obtained for different initial gap distances between the piezoelectric beam and the stopper. In addition, the bifurcation diagram of period solutions with a constant excitation frequency and varying gap distance was also attained.

Analytical Bifurcation Tree of Period-1 to Period-4 Motions in a 3-D Brushless DC Motor With Voltage Disturbance

In this paper, the bifurcation trees of analytical solutions of a 3-D brushless DC motor with the voltage disturbance are obtained through the generalized harmonic balance method. The electrical and mechanical model of the 3-D brushless motor is transformed to the dynamic system of coefficients of finite Fourier series. Stable and unstable analytical solutions of the 3-D brushless motor are solved based on such a Fourier series coefficient system. Bifurcation trees of analytical solutions of period-1 to period-2 and period-1 to period-4 motions are achieved. Stability and bifurcations of the analytical solutions of the 3-D brushless motor are determined by the eigenvalues of Jacobian matrix of the coefficient dynamic system. Frequency-amplitude characteristics of periodic motions are presented for a better understanding of the motion complexity in frequency domain. Numerical illustrations are completed for comparison of the analytical solutions with numerical results. The complex dynamics of the 3-D brushless motor are exhibited through the bifurcation trees of analytical solutions.

An independent period-3 motion to chaos in a nonlinear flexible rotor system

In this paper, a bifurcation tree of an independent period-3 motion to chaos in a flexible nonlinear rotor system is developed semi-analytically. The period-3 motion was traditionally called the subharmonic periodic motion of order-1/3, but one did not achieve the corresponding solutions yet. Herein, stable and unstable periodic solutions on the bifurcation tree in the flexible rotor system are achieved and the corresponding stability is analyzed by eigenvalue analysis. Harmonic frequency-amplitude characteristics for periodic motions on the bifurcation tree are presented. For comparison of analytical and numerical results, numerical simulation of periodic motions is completed. Phase trajectories, displacement orbits and velocity planes are illustrated, and harmonic amplitude spectrums of period-3 and period-6 motions are presented to show harmonic terms effects. Such studies presented in this paper help one better understand the subharmonic periodic oscillation of order-1/3 in nonlinear rotor systems.

On a Global Sequential Scenario of Bifurcation Trees to Chaos in a First-Order, Periodically Excited, Time-Delayed System

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.

Frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator

In this paper, nonlinear frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator are studied systematically. The periodic motions of the van der Pol oscillator are determined by the semi-analytical method, and the corresponding stability and bifurcation analysis is completed through the eigenvalue analysis. From the finite Fourier series analysis, the nonlinear frequency-amplitude characteristics of periodic motions are analyzed. From the frequency-amplitude analysis, the limit cycle of the van der Pol oscillator can be obtained analytically as excitation amplitude approach to zero, rather than numerically. For the van der Pol oscillator, most of periodic motions in the van der pol oscillator are symmetric. However, an asymmetric period-1 motion in the van der Pol oscillator is discovered. Thus a bifurcation tree of period-1 motion to chaos can be found.

Periodic motions to chaos in a 1-dimensional, time-delay, nonlinear system

In this paper, periodic motions varying with excitation strength in a 1-dimensional, time-delay, nonlinear dynamical system are studied through a semi-analytical method. With varying excitation strength, a global order of bifurcation trees of periodic motions is given by G1(S) ◁ G1(A) ◁ G3(S) ◁ G2(A) … ◁ Gm(A) ◁ G2m+1(S)◁ … (m = 1,2, …) where Gm(A) is for the bifurcation tree of asymmetric period-m motions to chaos, and G2m+1(S) is for the bifurcation tree of symmetric period-(2m + 1) motions to chaos. On the global bifurcation scenario, periodic motions are determined through specific mapping structures, and the corresponding stability and bifurcation of periodic motions are determined by eigenvalue analysis. Numerical simulations of periodic motions are carried out to verify analytical predictions. Phase trajectories and harmonic amplitudes of periodic motions are presented for a better understanding of the 1-dimensional time-delay system. Even for weak excitation, the traditional methods still cannot be applied to such a time-delay nonlinear system.

On bifurcation trees of period-1 to period-2 motions in a nonlinear Jeffcott rotor system

In this paper, periodic motions in a nonlinear rotor system are predicted through an implicit mapping method. The bifurcation trees of different period-1 to period-2 motion are presented, and there are many segments for different period-1 and period-2 motions. Harmonic frequency-amplitude characteristics of periodic motions are discussed for analysis and design of nonlinear rotor systems. Numerical simulations of periodic motions are completed for comparison of predicted and numerical results. The predictions of periodic motions in the nonlinear rotor system can provide guides for practical measurements of rotor systems, and one can have a better understanding of nonlinear rotor dynamics compared to the traditional perturbation analysis. Such studies presented in this paper just throw out some guides and ideas to predict complex motions in nonlinear rotor systems. The authors hope one can follow what presented in this paper to do further studies in nonlinear rotor dynamical systems and to improve design and control of nonlinear rotor systems.

Correction to: Bifurcation trees of period-1 motions in a periodically excited, softening Duffing oscillator with time-delay

In the original publication, Fig. 1a, b was published incorrectly. The corrected figure is given below.

On period-1 motions to chaos in a 1-dimensional, time-delay, nonlinear system

In this paper, bifurcation trees of period-1 motions to chaos in a 1-dimensional, time-delay, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motions. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation trees of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

Nonlinear models and bifurcation trees in quantum mechanics: a review of recent results

In this talk we discuss some recent results I obtained for a class of nonlinear models in quantum mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark–Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs; for a detailed treatment we refer to Sacchetti (Phys Rev E 95:062212, 2017, SIAM J Math Anal 50(6):5783–5810, 2018) where this model has been studied.

Influence of Collaterals on True FFR Prediction for a Left Main Stenosis with Concomitant Lesions: An In Vitro Study.

With the aim of assisting interventional cardiologists during decision making for revascularization, reduced-order (0D) approaches have been developed to predict the true fractional flow reserve (FFRTrue) of individual stenoses in multiple-lesion arrangements. In this study, a general equation was derived to predict the FFRTrue of a left main (LM) coronary stenosis with downstream lesions, one in the left anterior descending (LAD) and the other in the left circumflex (LCx) artery, and distinct collateral circulations supplying each daughter artery. An in vitro model mimicking the fractal nature of LM bifurcation trees with collateral branches was developed to validate the FFR values obtained with the prediction model (FFR Pred Model ). Our results demonstrated that: (1) considering collaterals significantly improved the FFR Pred Model estimation for a moderate LM stenosis with two downstream lesions as compared to computations with no collateral consideration (p < 0.001): mean absolute error |FFR Pred Model - FFRTrue| ± SD was equal to 0.02 ± 0.01 vs. 0.04 ± 0.02 respectively, and (2) Deviations from FFRTrue for LM stenoses are correlated to both, downstream lesion severities and collateral developments. The present study supports the hypothesis that collateral circulations supplying the LAD and LCx must be considered when predicting the FFRTrue of an LM stenosis with downstream lesions.